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For $G$ a group and

$$H \xhookrightarrow{\iota} G$$

a subgroup, write

(1)$$\iota^\ast \;\colon\; G Rep_{\mathbb{C}} \xrightarrow{\;} H Rep_{\mathbb{C}}$$

for the functor of restricted representations.

Proposition 0.1. The functor $\iota^\ast$ (1) has a right adjoint $\iota_\ast \colon H Rep_{\mathbb{C}} \xrightarrow{\;} G Rep_{\mathbb{C}}$ given by

$$V \in H Rep_{\mathbb{C}} \;\;\;\;\;\; \vdash \;\;\;\;\;\; \iota_{\ast} V \;\coloneqq\; Hom_H\big( \mathbb{C}[H] ,\, V \big) \,,$$

where on the right we have the vector space of $H$-equivariant linear maps equipped with the $G$-group action given by

$$\left. \begin{array}{l} f \,\colon\, \mathbb{C}[H] \xrightarrow{\;} V \\ g \,\in\, G \end{array} \right\} \;\;\;\;\;\; \vdash \;\;\;\;\;\; g \cdot f \,\coloneqq\, f(- \cdot g) \,.$$

Proof. We claim that the hom-isomorphism is given by evaluation at the neutral element $\mathrm{e} \in G$:

$$\left. \begin{array}{l} V \,\in\, H Rep \\ W \,\in\, G Rep \end{array} \right\} \;\;\;\;\;\;\;\; \vdash \;\;\;\;\;\;\;\; \frac{ W \xrightarrow{\;\;\;\; f \;\;\;\;} \overset{\iota_\ast V}{ \overbrace{ Hom_{H}\big( \mathbb{C}[H] ,\, V \big) } } }{ \iota^\ast W \xrightarrow{\;\; f(-)(\mathrm{e}) \;\;} V }$$

To see this, just observe that

$$\left. \begin{array}{l} f \,\in\, Hom_G\Big( W ,\, Hom_H\big( \mathbb{C}[H] ,\, V \big) \Big) \\ h \,\in\, H \end{array} \right\} \;\;\;\;\;\;\; \vdash \;\;\;\;\;\;\; f(h \cdot -)(\mathrm{e}) \;=\; \big( h \cdot f(-)\big)(\mathrm{e}) \;=\; f(-)(\mathrm{e} \cdot h) \;=\; f(-)(h) \,,$$

where the first equality is the $G$-equivariance of $f$, and the second is the $H$-equivariance of $f(-)$. This shows that $f(-)(\mathrm{e})$ is $H$-equivariant and that it uniquely determines $f$.

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On electrodynamics via field strengths instead of gauge potentials:

• Joshua Newey, John Terning, Christopher B. Verhaaren: Geometrizing the Anomaly [arXiv:2504.16998]